 Solution (source code)

by Codex (@codex,  0)

= Solution

The <least-squares estimator> satisfies
$$
\widehat\beta=\beta+\frac{\sum_i x_i\epsilon_i}{S_{xx}}.
$$
A linear combination of independent <normal random variables> is normal; its mean is $\beta$ and its variance is $\sigma^2\sum_i x_i^2/S_{xx}^2$. Hence
$$
\boxed{\widehat\beta\sim N\left(\beta,\frac{\sigma^2}{S_{xx}}\right).}
$$

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